THE DYNAMICS OF CAPITAL ACCUMULATION IN THE US: SIMULATIONS AFTER PIKETTY
By
Philippe De Donder and John E. Roemer
April 2015
COWLES FOUNDATION DISCUSSION PAPER NO. 1998
COWLES FOUNDATION FOR RESEARCH IN ECONOMICS YALE UNIVERSITY Box 208281 New Haven, Connecticut 06520-8281
http://cowles.econ.yale.edu/ 16 April 2015
“The dynamics of capital accumulation in the US: Simulations after Piketty” by † * Philippe De Donder and John E. Roemer
Abstract: We calibrate a sequence of four nested models to study the dynamics of wealth accumulation. Individuals maximize a utility function whose arguments are consumption and investment. They desire to accumulate wealth for its own sake – this is not a life-cycle model. A competitive firm produces a single good from labor and capital; the rate of return to capital and the wage rate are market-clearing. The second model introduces political lobbying by the wealthy, whose purpose is to reduce the tax rate on capital income. The third model introduces differential rates of return to capitals of different sizes. The fourth model introduces inheritance and intergenerational mobility.
Key Words: Piketty, dynamics of wealth accumulation, intergenerational mobility, Kantian equilibrium JEL codes: D31, D58, E37
† Toulouse School of Economics (GREMAQ-CNRS and IDEI), [email protected] * Yale University, [email protected] 1
1. Introduction We simulate the dynamics of wealth accumulation in a model calibrated to the US economy, with initial conditions taken from 2012 data. Although the model is inspired by Piketty (2014), we have added quasi-traditional micro-foundations: at each date there is an equilibrium where the return to capital and the real wage balance the markets for capital and labor. The equilibrium is only ‘quasi’-traditional for the following reason. Agents do not maximize a discounted infinite sum of consumption over time, or even of consumption over time and a bequest, but rather a single-period utility function of consumption and savings, repeated in each period. That is, agents desire to consume and to accumulate wealth for its own sake. This assumption is inspired by Marx, who argued that, while pre-capitalist economies produced commodities, exchanging them for money in order to purchase a different commodity bundle (C − M − C′ , à la Marx) , capitalist property relations produce agents whose aim is to transform money into more money, via the intermediary of commodities (labor power and capital), thus M − C − M ′ . Piketty (2014) argues – effectively, we believe – that the life cycle model, which is the C − M − C′ formulation – is not supported by the data. Moreover, our belief is that the very wealthy do not accumulate in order to leave bequests. They do so because ‘money is life’s report card,’ as the caption of a New Yorker cartoon said. Warren Buffet (Andrew Carnegie) does not (did not) accumulate wealth in order to enrich his children; nor were his motives primarily philanthropic, although Buffet will (Carnegie did) use his winnings in the capitalist game for philanthropic ends. Accumulation for its own sake is the motivation for most members of the wealthy class, for success in the game of life is judged by one’s wealth, often signaled by one’s consumption. We begin by postulating a distribution of wealth given by the 2012 US wealth distribution, from Piketty and Zucman (2013). We postulate a lognormal distribution of skills. We assume that the distribution of wealth is monotone increasing in an individual’s skill. A firm, using a CES production function whose inputs are efficiency units of labor and capital, maximizes profits. Consumer-workers offer inelastically their entire endowment of skilled labor to the firm; they demand the consumption good and supply capital to the firm in order to maximize preferences described above. The interest rate and real wage equilibrate the markets for labor and capital. There are proportional taxes on capital and labor income, the revenues from which are returned as a demogrant to each worker. In the basic model these 2
tax rates are the same, and an equilibrium is re-established at each period. The main fundamental changing over time is the distribution of capital/wealth; there is also an exogenously growing distribution of skills, and of the ‘expected’ standard of living. We introduce sequentially three more models. In Model 2, we append to the basic model lobbying by the wealthy, who contribute to a fund that is spent to convince legislators to reduce the tax rate on capital income. We describe below how the wealthy overcome the free rider problem to generate a significant lobbying fund. Consequently, the tax rate on capital income now declines, while the tax rate on labor income is exogenous and fixed. We next amend Model 2 by introducing differential rates of return to capital: those with large amounts of capital can attain a higher rate of return than those with small amounts. This is, again, an important fact discussed by Piketty (2014). In Model 3, this amendment accounts for a more dramatic concentration of capital than occurred in the first two models. Finally, in Model 4, we add to Model 3 intergenerational mobility. We take a generation to last for 50 (or 25) years, and model this by assuming each individual has a probability of 2% (or 4%) of dying each year, upon which his capital passes down, without taxation, to his single offspring. The offspring’s skill level – and hence her labor earnings – are not inherited, but are taken to be determined by the income intergenerational mobility matrix of Chetty et al (2014). The population dynamics are made explicit below. The dynamics of wealth again change. Table 1 summarizes the main characteristics of the four nested models.
Model number Lobbying Differential rates of Intergenerational return to capital Mobility 1 no no no 2 yes no no 3 yes yes no 4 yes yes yes Table 1: Four nested models
3
We present the simplest model (Model 1) in section 2, with sections 3 to 5 devoted to Models 2 to 4. In each section, we first develop analytically the model, calibrate it and then provide and comment on results from numerical simulations. Section 6 concludes.
2. Model 1: Basic
We start by describing analytically the basic model developed, before calibrating it and presenting our numerical results.
A. Analytical description of the model
i. Endowments There is a distribution of skill, s, denoted by a distribution function F on [0,∞) where ssdFs= ∫ () denotes the average skill. There is a distribution of wealth at date 0 given by a function S0 (s) . It is postulated that wealth/capital is monotone increasing in s. We assume that skills increase (exogenously) by a factor of (1 + g) per annum.
ii. Production There is a single good produced by a single firm, using a CES technology given by y(K, L) = A(aK (δ−1)/δ + (1 − a)L(δ−1)/δ )δ/(δ−1) , (2.1) where y , K and L are per capita income, capital, and labor in efficiency units. The only technical change in the model is induced by the exogenous increase in labor skills. The firm faces an interest rate r and a real wage per efficiency unit of labor w, and maximizes profits.
Consequently, the demands for labor and capital by the firm at date t are determined by the
first-order conditions from maximizing profits defined as y(Kt , Lt ) − wt Lt − (rt + d)Kt :
1/δ−δδ−δ 1/ ( 1)/ 1/ δ−δδ−δ 1/ ( 1)/ rdayKttt+= A and w t = (1− ayL ) tt A , (2.2) where d is the annual rate of capital depreciation. Note that the firm replaces depreciated capital from income. Economic profits are zero. 4
iii. Preferences Preferences are non-traditional. We assume there is a socially expected standard of
living for people in this society, produced by a consumption level c0 at date 0. This expected consumption level increases by a factor of (1 + g) per annum. We do not call this subsistence consumption – we set it at $100,000 in the simulations. It is the consumption level to which ordinary people aspire, which is generated by advertising and the media. (In the US, this level would define a successful middle-class life.) A sufficiently wealthy individual at date t chooses her consumption c and investment I to maximize a Stone-Geary utility function as follows:
t−1 α 1−α max(ct − c0 (1 + g) ) It
s.t. ct + It ≤ yt (s) (2.3) t−1 ct ≥ c0 (1 + g)
where yt (s) is the income of individual s at date t. If there is no solution to program (2.3), t−1 because income is insufficient to purchase the consumption level c0 (1 + g) , then the individual consumes out of wealth. To be precise:
cst ()= ⎧ (2.4) ys()++ S (), s if ys () S () s≤ (1+ g )t−1 c (case 1) ⎪ tt−−110 tt ⎨ tt−−11 (1+ gc )001 , if ysttt ( )≤ (1+ gc )≤ ysSs ( )+ − ( ) (case 2) ⎪ ttt−−−111 ⎩⎪(1++gc )000α− ( ystt ( ) (1+>+ gc ) ), if ys ( ) (1 gc ) (case 3)
In case 1, the individual consumes his income plus his wealth Sst−1(), and those together do not suffice to generate the socially expected consumption of (2.4). In case 2, when her income does not suffice to allow socially expected consumption but her total asset position does, she consumes exactly socially acceptable consumption. In case 3, where her income alone suffices to allow socially acceptable consumption, she solves program (2.3) with α the marginal propensity to consume out of income. Thus, investment is given by: Is()= ⎧ . (2.5) t −Ss( )< 0, if case 1 ⎪ t−1 ⎨ys( )− c (1+< g )t−1 0, if case 2 ⎪ t 0 ⎩⎪ ystt()− cs ()> 0, if case 3 The dynamics of wealth are given by: 5
St (s) = St−1(s) + It (s) . (2.6) Note that, because the firm replaces depreciated capital from its income, the investor can cash out his entire capital stock at the end of the period, and so the depreciation does not appear in equation (2.6).
(iv) Equilibrium The market clearing equations are: ∞∞ K==+=+ S() sdFs (), L (1 g )tt−−11 sdFs () (1 g ) s . (2.7) tt∫∫−1 t 00
(v) Income We assume an exogenously given income tax rate τ , the revenues from which are returned to citizens as a demogrant. Thus income for an agent of type s in year t is given by:
tt−−11 ysttttttt( )= (1−τ )( wsgrSswsgrK (1++ )−1 ( )) +τ ( (1++ ) ). (2.8)
(vi) Summary of equations and solving of the model The equations summarizing the model are : 1. FOC, profit maximization w.r.t. K: /(1 ) δ−1 δ−δ ⎛⎞⎛⎞rd+ ⎜⎟t − a ⎜⎟aA Kgs=+⎜⎟⎝⎠ (1 ) t−1 (2.9) t ⎜⎟1− a ⎜⎟ ⎝⎠ 2. FOC, profit maximization w.r.t. L:
(δ−1)/δ 1/(δ−1) ⎛ ⎛ K ⎞ ⎞ w (1 a)Aa t 1 a (2.10) t = − ⎜ ⎜ t−1 ⎟ + − ⎟ ⎝ ⎝ (1 + g) s ⎠ ⎠ 3. Definition of consumption: see equation (2.4) 4. Definition of investment: see equation (2.5) 5. Intergenerational transmission of wealth: see equation (2.6) 6. Market clearing of capital and labor markets: see equations (2.7) 6
7. Definition of income: see equation (2.8)
The full model is solved as follows, beginning with the wealth function St−1(s) at the beginning of date t. Kt and Lt are determined by (2.7). Equations (2.9) and (2.10) determine rt and wt . yt (s) is determined by (2.8). Consumption and investment are determined by (2.4) and (2.5). St (⋅) is determined by (2.6). The next iteration begins.
B. Calibration of the model We describe our calibration of preferences, the production function, and the initial distribution of capital.
(i) Preferences
One period is deemed to be one calendar year. We choose c0 = 100 (thousands of dollars), which grows at a factor of (1 + g) each year, with g=0.01. We choose α = 0.6 , based on the fact that the propensity to consume for the wealthy is about 0.6 out of income.
(ii) Production function We assume a CES production function given by (2.1). Piketty (2014) argues that a choice of the elasticity of substitution δ∈(1.3,1.6) is implied by the historical distribution of 1 capital’s share in income. We choose δ = 1.5 . The capital income ratio Ky/ is 4.5 in the U.S. Depreciation is about 10% of GNP, which suggests a rate of depreciation d = 0.02. Using (2.2), calculate that : (1/δ− ) 1 ()rdK+ ⎛⎞ y = a⎜⎟ , (2.11) yAK⎝⎠ and hence 11/−δ ⎛⎞rK K⎛⎞ y ad=+⎜⎟⎜⎟ . (2.12) ⎝⎠yyAK⎝⎠
1 Karabarbounis and Neiman (2014) estimate an elasticity of substitution around 1.25 from cross-country variation in trends in rental rates and labor shares. 7
K rK Taking = 4.5 and capital’s share in income = 0.28 (both from Piketty (2014)), (2.12) y y reduces to a = (0.28 + 0.09)(4.5)−.333 A−.333 . (2.13) From the production function, we have: 1((/y)(1)(/y))=+AaK.333− a L .333 3 . (2.14)
The distribution Fs() of skills is taken to be lognormal. The unit of skill has no meaning: we take the median skill level to be 0.85 and the mean s to be 1. We then have that
L=1 at period 1. Equation (2.14) then becomes 1=+AaK ( ( / y).333 (1− a )( L / y) .333 ) 3=+ Aa ( (4.5) .333 (1− a )(108.3)− .333 ) 3 , (2.15) where we calculate L / y= (108.3)−1 , using the facts that total income is 16.8× 109 in thousands of dollars and the size of the labor force is 155× 106 . (Thus, income per worker is $108,300.) Solving (2.13) and (2.14) simultaneously for (a, A) , we have (a, A) = (0.070,33.41) .
(iii) The initial distribution of capital, S0 (s) The initial distribution of wealth is taken from Saez and Zucman (2014), Appendix 2 Table B1: Top wealth shares. We assume that the wealth distribution is linear by parts over s, with 7 different brackets. The first bracket corresponds to no capital at all for the bottom half of the skill/wealth distributions. The remaining 6 brackets reproduce the top wealth shares reported by Saez and Zucman (2014) for 2012 (the most recent year for which data is available): 77.2% for the top 10%, 64.6% for the top 5%, 41.8% for the top 1%, 34.5% for the top 0.5%, 22% for the top 0.1% and 11.2% for the top 0.01%. Total capital at the beginning of period 1 is given by (ii) above (4.5 times $108,300). The taxation rate τ is set at 0.35 throughout the paper.
2 Available at http://gabriel-zucman.eu/files/uswealth/AppendixTables(Distributions).xlsx 8
C. Numerical results
We report in Table 1 the main results obtained from solving the model numerically from t=1 to t=25 (first column). The second column shows that the equilibrium interest rate r is fairly stable and decreases from 6.2% to 5.7% over 25 years. The equilibrium wage rate w increases from $68,240 to $71,014 over the same period. GDP per capita increases from $108,740 to $146,580. It is clear from Table 1 that both the growth rate of GDP and of average wealth is much lower than the equilibrium interest rate. So, in Piketty (2014)’s parlance, we obtain that g Insert Table 1 around here The last 5 columns of Table 1 document the evolution of the wealth distribution over 25 years. The bottom half of the population does not accumulate any capital over that period. The patrimonial middle class (defined as agents in-between the fifth and ninth deciles in the skill/income/wealth distribution) sees its share of capital decrease from 22% to 17%, while the share of the 90th to 99th percentiles increases from 35.8% to 40%. The share of the top percentile then increases very slightly (from 42.1% to 42.8%), but the top 0.1% sees its share remaining fairly stable around 22%. Figure 1 complements this information by comparing the distribution of capital as a function of skill in periods 1 (dashed line) and 25 (thick line). We first see that people immediately above the median skill (s=0.85) end up consuming all their capital, so that the share of people without any capital increases from 50% at time 1 (our calibration assumption) to 69.5% of the population at t=25 (corresponding to s=1.13). Individuals situated between the 69.5th and 72.7th percentiles have less capital (in absolute amount) at time 25 than at time 1, while the top 27% of the distribution sees its capital grow in absolute amounts. The wealth distribution remains linear by part following our calibration assumptions. The three 9 vertical dashed lines correspond to the 50th, 90th and 99th percentiles of the skill distribution, respectively. Insert Figure 1 around here Figure 2 shows the density of the wealth distribution at time t=25 as a function of skills s –i.e., it interacts the function S(s, 25) with the distribution of skills F(s). The vertical dashed lines correspond to the 50th, 90th and 99th percentiles of the skill distribution. We see that the density is inversely U-shaped for the three groups spanned by these dashed lines: the 50-90, 90-99 and 99+. The density reaches its overall maximum for agents situated slightly above 90% in the distribution. Insert Figure 2 around here We postulate a distribution of skills that is lognormal. However, we know that the income distribution is well-approximated by a distribution that is lognormal for an interval of incomes up to a quite large income y* , and a Pareto distribution on the income interval y ≥ y* . We wish to see how well the distribution of income generated by our simulations tracks this kind of distribution. Jantzen and Volpert (2012) characterize the Lorenz curve of a hybrid income distribution, which exhibits left-sided self-similarity (where the degree of inequality repeats itself as we restrict ourselves to poorer and poorer fractions of the population) up to a threshold income level, and right-sided self-similarity (i.e., a Pareto distribution) above that threshold. Interestingly, the equation of the Lorenz curve is of the form: L(x) = xa (1 − (1 − x)b ) , (2.16) where a > −1 and b > 0 ; surprisingly, the Lorenz function has only two parameters. These authors fit the empirical income distributions of Atkinson, Piketty and Saez (2011), choosing parameters (a,b) by least-squares, and achieve remarkably good fits. We fit our simulated income distributions using least-squares and (2.16). 10 The result is depicted in Figure 3, where the equilibrium income distribution at period t=25 is shown using dots while the fitted distribution (corresponding to a=-0.28 and b=0.37 in (2.16)) is represented by a continuous blue curve. We obtain what seems to be an excellent fit, with a R square of 0.932, although our numerical computations slightly underestimate the total income shares at both the bottom and the top of the distribution, compared to the fitted distribution. The Gini coefficient corresponding to the fitted Lorenz curve is 0.386. Insert Figure 3 around here To better judge how well our equilibrium income distribution fits this Lorenz curve, we compare our distribution with the Lorenz curve obtained from Jantzen and Volpert (2012) without the Pareto tail. They show that this curve is of the form: Lx()= cxp , (2.17) with two parameters c and p, which we choose by least-squares. Although we have two degrees of freedom when fitting our distribution to this family of Lorenz curves, as previously, we achieve a worse fit, with R2=0.9084. Figure 3 reports (in red) the best-fitted Lorenz curve without a Pareto tail. It is clear that, even though we assume a simple lognormal distribution of skills, our equilibrium income distribution after 25 periods is much closer to the hybrid distribution from Jantzen and Volpert (2012) (which they claim represents especially well the US income distribution) than to the one generated from their model without a Pareto tail. We have also checked the robustness of our numerical results to the introduction of a Pareto tail in the skill distribution. We detail in Appendix how we have proceeded for the calibration of this hybrid skill distribution. The numerical results we obtain are very similar to the ones reported above – see Table xx in the Appendix, and compare it to Table 1 above. The figures corresponding to Figures 2 and 3 are also very similar and are omitted for the sake of brevity. Our results are then robust to the introduction of a Pareto tail in the skill distribution. We have also fitted the new equilibrium income distribution to the Jantzen and Volpert (2012) family of Lorenz curve, and we obtain a slightly better fit, as measure by the R2 which 11 increases from 0.932 to 0.955. The Gini coefficient associated with this fitted Lorenz curve increases slightly, from 0.386 to 0.391. To summarize the results obtained with Model 1, we observe that the growth rate of income and of average wealth are significantly smaller than the interest rate (g We now introduce lobbying by the wealthy into the picture, and assess its impact on the equilibrium dynamics of our model. 3. Model 2: Lobbying A. Analytical model Formally, the purpose of lobbying by the wealthy is to reduce the tax rate on capital income. The model will henceforth contain two tax rates, τ on labor income and τc on capital income. If the per capita (in the whole population) expenditure on lobbying at time t c is σt , then the tax rate τt is defined by: (σ /(1+ g)t−1)β τ−τc = k t (3.1) t β where k and β are parameters to be specified below. The effective cost of achieving a given outcome (tax rate τc ) grows at rate g. 12 However, our conception of ‘lobbying’ is in reality much richer: it includes all those expenditures by owners of capital to protect their wealth from confiscation, which may occur not only through taxation but through other means. These expenditures include those needed to create an ideology among voters which is pro-capitalist and pro-laissez-faire, what has been called in recent years the neoliberal worldview. Thus, it includes expenditures on conservative think tanks, and on the transmission of this ideology through the media. It includes what Samuel Bowles has called ‘guard labor,’ labor needed to protect property from theft. Bowles and Jayadev (2014) argue that the amount of guard labor in the advanced capitalist countries is highly correlated with their Gini coefficients; they estimate that in the US in 2000, there where 200 workers employed in guard labor per 10,000 workers, the highest ratio of the advanced countries. (They include in guard labor, besides police and security guards, the military, prison officials, and weapons producers.) ‘Lobbying’ must also include those expenditures to prevent confiscation of profits by workers through labor organization; thus, the expenditures of the capitalist class in preventing the formation of unions, or destroying existing unions, must be included as well. Union-busting has been important in the United States during the last forty years, when the density of unionized workers has fallen from around 30% in 1960 to less than 10% today. (To see this is not simply an effect of structural changes in production, note that union density in Canada is roughly still 30% today as it was in 1960.) We would include in ‘lobbying expenditures’ corporate legal fees that are used to protect capital from confiscation by governments and workers, but not intra-capitalist transfers within a country, which do not change (to a first approximation) the ownership of capital by classes. How is the budget for ‘lobbying’ raised in our model? The reduction in the tax rate on capital income is a public good for the wealthy, and so there is a free-rider problem in generating contributions to that budget under the usual assumptions about behavior. We solve the problem by assuming that those who contribute to lobbying optimize in the Kantian manner: the lobbying equilibrium is a multiplicative Kantian equilibrium of the game among contributors. (See Roemer (2014) for a discussion of Kantian equilibrium; see Roemer (2006) for a political model where the determination of citizen contributions to politics is modeled as a Kantian equilibrium. ) In public-good games, the multiplicative Kantian equilibrium leads to a Pareto efficient solution of the public-good problem . 13 Let the lobbying expenditure of an agent of type s at date t be σt (s) . The individual’s income at any contribution σ is: tc−−11 tc ystt(σ ; )= (1−τ ) wsg (1++ ) (1−τ ( σ )) rSswsg tttttt−1 ( )+ τ (1++ )τσ ( ) rK −σ (3.2) where σ = σ (s)dF(s) . The equations (3.2) define payoff functions of a game for the ∫ t citizens; a multiplicative Kantian equilibrium is a function σ(⋅) such that no player would advocate that all players change their contributions by any factor (the factor being the same for the entire society). The FOC for the solution of this problem is a schedule of contributions σ(⋅) satisfying : dy(();)ρσ s s d (1cc ( ))rS ( s ) ( ) rK ( s ) 0 . (3.3) = ()−τ ρσ tt−1 + τρσ t t −ρσ = ddρρρ=11ρ= Expanding this expression gives: c′ σ(s) = max[−(τ (σ))rt σ(St−1(s) − Kt ),0] , (3.4) where τc′ is the derivative of the function τc (σ) implicitly defined by equation (3.1). It follows that positive contributions to lobbying are made by precisely those agents for whom St−1(s) ≥ Kt -- that is, those whose capital is at least equal to the average capital endowment * in the society. Thus, those who contribute to the lobbying effort are exactly {s ≥ st } where * * st is defined by: St−1(st ) = Kt . (3.5) We have by definition: ∞ σ = σ (s)dF(s) . (3.6) t ∫ t * st Integrating equation (3.4) and dividing by σ , we have: ∞ 1()(())()= −τc′ σrSsKdFs − (3.7) tt∫ t−1 t * st c c from which we solve for σt and τt = τ (σt ). Finally, we solve for the function σt (⋅) using (3.4). A critic might well say that we should solve for the Kantian equilibrium of lobbying contributions not by having the investor maximize his income, but rather his utility. We have chosen not to do so, because when the investor maximizes income, we have a simple analytic 14 * solution for st (from (3.5)) and this simplifies the simulations considerably. This would not be the case were the investor to maximize her utility. The equations summarizing the model are the same 7 equations given in subsection 2.A. (vi), where the definition of income is now given by (3.2). We add the following four equations to these seven: 8. Lobbying contribution of a type s investor σt ()s = ⎧ (3.8) ⎪ 0, if ss≤ * ⎨ t ⎪ c′ * ⎩−τ() σtttrSsK σ ( t−1 () − t ), if ss> t 9. Determination of per capita lobbying expenditures ⎛⎞∞ ()σ 1(1)−β=+kg (1) −t β ⎜⎟ rSsKdFs ()− () (3.9) tttt∫ ()−1 ⎜⎟* ⎝⎠st 10. Effect of lobbying on taxation of capital income β ⎛ σ ⎞ ⎝⎜ (1 + g)t−1 ⎠⎟ τc (σ) = τ−k (3.10) β 12. Definition of lowest type that contributes to lobbying: see equation (3.5) As for solving the model, we proceed as previously to obtain Kt and Lt , and then rt and wt . σt is determined by equation (3.9), and σt (s) is determined by equations (3.8) and (3.5). c τt is determined by (3.10). yt (s) is determined by (3.2). Consumption, investment and saving at the end of period t are then obtained as previously, and we start the next iteration. B. Calibration The tax rate on labor income is maintained at τ = 0.35 , as previously. To calibrate (3.11), we choose (k,β) = (0.06, 0.40) . We do not have a justification. 15 C. Numerical results The evolution of our toy economy from t=1 to t=25 is very similar in Models 1 and 2, with the same trends in all variables. We then assess the impact of lobbying on the equilibrium by comparing the last row of Tables 1 and 2. The lobbying activity results in more than halving of the capital income tax rate (from 35% in Model 1 to 15%). This results in a slightly larger GDP after 25 periods ($148,640 per capita, compared to $146,580 without lobbying) and a slightly higher capital output ratio (5.08 against 4.97 at t=25). The equilibrium interest rate is slightly lower (5.62% against 5.74%) while the equilibrium wage rate is slightly higher ($71,680 compared to $71,015). The share of capital (including depreciation) in total income increases very slightly to 38.8%. Insert Table 2 around here At equilibrium, the top 17% of the income/wealth distribution contribute to lobbying (a proportion slightly decreasing with time) and their per capita (in the whole population) contribution increases from $1,489 at t=1 to 2 604$ at t=25, representing 1.75% of GDP. This increase is faster than the exogenous growth rate g = 1%, resulting in an equilibrium capital income tax decreasing from 17.4% at t =1 to 15% after 25 periods. In terms of wealth distribution, the opportunity to lobby benefits the top 10% (and especially the top 1%) at the expense of the rest of the distribution: the patrimonial middle class (50th to 90th percentile) sees its wealth share decrease to 15.6% at t=25 (compared to 16.9% without lobbying). The 90th to 99th percentile see their share increase by 0.4% thanks to lobbying (from 40.2% to 40.6%), the top 1% by 1% (from 42.8% to 43.8%) and the top 0.1% by 0.6% (from 21.8% to 22.4%). For the latter group, we now obtain that their wealth 3 share increases with time, rather than decreasing without lobbying. The fitted Lorenz curve still looks quite close to the obtained distribution, with parameters a=-0.25 and b=0.33), although our numerical computations still slightly 3 We obtain the same trends for all variables with the hybrid distribution of skills described in the Appendix, except that the top 0.1% wealth share remains decreasing with time. 16 underestimate the total income shares at both the bottom and the top of the distribution. The fit is actually better than in Model 1, with an R square increasing to 0.95. The fitted income distribution of Model 1 Lorenz dominates the one of Model 2, and the Gini coefficient 4 increases from 0.386 to 0.441. The figures depicting wealth as a function of skill, the density of wealth distribution, and the Lorenz curves are sufficiently close to those presented in the previous section that we delay until the end of the paper the comparison of the curves for the four models. We now summarize the impact of introducing lobbying. Lobbying results in a sizeable decrease of the capital income tax rate which generates a further capital deepening of the economy, measured by the capital output ratio. But the main impact of lobbying is distributional. The wealth distribution becomes even more unequal, with the shares of the top 10%, top 1% and top 0.1% increasing at the expense of the patrimonial middle class. Income inequality increases too, with the new income distribution being Lorenz dominated by the one generated by Model 1, increasing the Gini coefficient. What is the magnitude of lobbying in the model? At t = 25 , lobbying expenditures are 1.7% of GDP. In 2012, $6.5 billion was spent on elections, $3.3 billion on congressional lobbying, and about $1 billion by think tanks, most of which are conservative. This adds up to about 0.07% of GDP. Nevertheless, given our more inclusive conception of what the protection of capital requires, we do not believe that our figure of 1.7% is an overestimate. It may well be an underestimate considering what is at stake. 4. Model 3: Differential rates of return on capital A. Analytical model and calibration 4 The fit is even better with the hybrid distribution and increases from 0.955 in Model 1 to 0.977 in Model 2. Introducing lobbying also increases the Gini coefficient with the hybrid skill distribution, from 0.391 to 0.448. 17 Large capitals earn significantly higher rates of return than small ones. Piketty (2014) estimates these rates of return using university endowments in the United States, which must publish their rates of return. He reports that Harvard, Princeton and Yale, with endowments much larger than $1 billion, earned an average rate of return of 10.2% over the thirty year period 1980-2010. Endowments somewhat higher than $1 billion but less than those of the top three earned 8.8%, endowments between $500 million and $1 billion, 7.8%, endowments between $100 and $500 million, 7.1%, and endowments less than $100 million, 6.2%. All these numbers are much greater than capital per worker, which is K ( )Y / L = (4.5)(16.8 ×1012 ) / (155.×106 ) = $490,000 . The average capital will earn much Y less than 6.1%, being invested by non-professionals in money-market funds, bank certificates of deposit, and mutual funds. We model this as follows. We assume that the rate of return available for the capital owned by an agent of type s is given by: r rsˆ ()=+t 1ν Fs ()240+ (5−ν ) Fs () , (4.1) tt4 () t where rt is the market-clearing interest rate and νt is a constant to be determined below. Thus the smallest capitals earn one-fourth the market-clearing rate, and the largest earn 1.5 times the market-clearing rate, about six times what the smallest capitals earn. The choice of functional form in (4.1) is ad hoc. It is chosen to guarantee: r • that the rates of return of t and 1.5r are achieved at the endpoints of the 4 t interval, and • to give a value νt in the interval (0,5) as the solution of the equation ∞ rˆ (s)S (s)dF(s) = r K . (4.2) ∫ t t−1 t t 0 From (4.2), capital income in the society aggregates to the market-clearing interest rate times the stock of capital. The interval in which νt lies according to the second bulleted property is 18 5 necessary to ensure that rˆt (⋅) is an increasing function, which is obviously desirable . As the reader will see below, the formulation of (4.1) gives the desired properties. The equations summarizing the model are the same 11 equations given at the end of subsection 3.A., where r(t) is replaced by rˆt (⋅) where necessary in the equations determining income ((3.2)), individual ((3.8)) and per capita ((3.9)) lobbying contributions. We add the following two equations to these eleven: 12. Rate of return for investor of type s: see equation (4.1) 13. Consistency of rates of return with competitive rate of return: see equation (4.2) As for solving the model, we proceed as previously to obtain Kt and Lt , and then rt and wt . Equations (4.1) and (4.2) determine νt and hence rˆt (s) . We then proceed as explained at the end of section 3.A to solve the other variables. B. Numerical results Introducing differentiated rates of return on capital leads to a smaller fraction of wealthy people lobbying (from 16.9% at t=25 in Model 2 to 13.4% here) but with a much higher per capita lobbying contribution (from $2,604 in Model 2 to $3,491 here, increasing from 1.75% of GDP to 2.36%), leading to a smaller value of the capital income tax rate (which decreases from 15% in Model 2 to 12.5% here). Both GDP and capital at t=25 decrease compared to Model 2 (GDP goes from $148,640 per worker to $147,960 while capital per worker decreases from $755,810 to $747,020). The capital/output ratio decreases slightly (from 5.08 to 5.05), resulting in a slightly larger interest rate (increasing from 5.62% to 5.66%) and a slightly smaller wage rate (decreasing from $71,678 to $71,461). The share of capital income (including depreciation) in total income barely changes, increasing from 38.77% to 38.68%. 5 To accomplish both of the bulleted properties for the function rˆt , it is necessary to make the function increase very rapidly for F(s) near one – hence the large exponent, 40. 19 Insert Table 5 around here Figure 4 depicts the interest rate as a function of the location of the individual in the wealth distribution at date t=25. The horizontal line represents the equilibrium average rate of return of 5.66%. More than 90% of the population faces a rate of return of less than 2% (recall that more than 50% of the population has no capital income anyway), and 98.7% of the population faces a rate of return which is lower than the average. The concentration of wealth among the very top earners explains the skewness of the distribution of rates of return. Insert Figure 4 around here This skewness has a very large impact on the equilibrium distribution of wealth. The share of the patrimonial middle class drops consequently, from 15.6% in Model 2 at t=25 to 9.9%. The share of the 90 to 99th percentiles also decreases (from 40.6% to 36.2%). The main beneficiaries of the introduction of differentiated rates of returns on capital are the top 1% (whose share increases from 43.8% to 53.9%) and the top 0.1% (whose share increases 6 from 22.4% to 28.5%). The fitted Lorenz curve still looks quite close to the obtained distribution, with parameters a=-0.47 and b=0.27). The fit looks even better for the very top income, at the expense of a larger under-estimation (respectively, over-estimation) of the bottom (resp., middle) income levels – see Figure 5. The overall fit decreases, compared to Model 2, with an R square decreasing from 0.95 to 0.9. The income distribution with Model 3 is not Lorenz dominated by those generated by Model 1 nor 2, but the Gini coefficient increases from 7 0.441 to 0.468. 6 We obtain the exact same trends when comparing numerical results obtained under Models 2 and 3 with the hybrid skill distribution described in the Appendix. 7 Similarly, the value of R square decreases from 0.978 (Model 2) to 0.918 (Model 3) with the hybrid skill distribution, while the Gini index increases from 0.448 to 0.474. 20 Insert Figure 5 around here To summarize, the introduction of differentiated rates of return to capital decreases the capital income tax rate, GDP and capital accumulation, but affects very little the equilibrium interest and wage rates. The main impact is indeed distributional, with the top 1% and top 0.1% increasing their wealth at the expense of the bottom 99%. Although there is no Lorenz domination relation between the income distributions generated by Models 2 and 3, we observe a further increase in the Gini coefficient. 5. Model 4: Intergenerational mobility A. Analytical model and calibration We now introduce death and inheritance. When an adult dies, we assume that his capital passes (untaxed) to his only child. However, the child will not in general have the skill/income capacity of the father. We use the 100 ×100 intergenerational income mobility 8 matrix of Chetty et al (2014) to model this process. An element pij of this matrix is the fraction of sons of fathers at the ith centile of the income distribution who have incomes at the jth centile of their cohort’s income distribution. Indeed, we assume that the matrix P = {pij } defines the mobility of skill, hence earned income. To describe the dynamics, let us first suppose that all fathers die at once at the beginning of the year. If an s father dies at the beginning of year t, his son inherits St−1(s) . The son will be economically active beginning in year t. The sons are distributed on the skill distribution F according to P. Denote by Qi the ith centile of F, comprising a small interval of skills. Let s ∈Qi and s′ ∈Q j . Then the ‘number’ of sons who inherit from fathers of 8 Online Data Table 1: National 100 by 100 transition matrix, available online at http://obs.rc.fas.harvard.edu/chetty/website/v2.1/online_data_tables.xls 21 skill s and end up at skill level s′ will be equal to 100 pij f (s) f (s′) . Integrating this over s′ ∈Q j gives 100 p f (s) f (s )ds = p f (s) , (5.1) ∫ ij ′ ′ ij Q j 1 because ∫ f (s′)dF(s′) = , which is the correct number of sons of s fathers who end up Q j 100 at Q j . If we add up these numbers over all j, we have: , (5.2) ∑ pij f (s) = f (s) j which is the total number of sons whose fathers were of skill s. Now let’s look at all fathers s ∈Qi . The number of sons who end up at some s ∈Q j will be 100 p f (s ) f (s)ds = p f (s ) . Summing over i, we have ′ ij ∫ ′ ij ′ Qi , (5.3) ∑ pij f (s′) = f (s′) i which is correct number of sons at s′ . Now let’s compute the total savings inherited by children when their fathers’ estates pass to them. We are interested in the total savings at a generic skill level s′ in the son’s generation. These savings come from fathers’ wealth at date t −1 -- so these are the savings at the beginning of date t for the sons, before they have augmented their savings with their own income (which will be with wages at skill level s′ ). The amount of inheritance at each value of father’s s will be the same. The average inheritance for sons at s′ in centile j will be: 100 100 p S (s) f (s)ds ≡ S* (s′) . (5.4) ∑ ij ∫ t−1 t−1 i=1 Qi We now integrate this over all sons: 100 100 100 ⎛ 100 ⎛ ⎞ ⎞ 100 p S (s) f (s) f (s′)ds ds′ = 100 p S (s) f (s)ds f (s′)ds′ = ∑∑ ij ∫ ∫ t−1 ∑ ∫ ⎜ ∑ ij ⎜ ∫ t−1 ⎟ ⎟ j i j ⎜ i ⎟ j=1 i=1 Q Q j=1 Q ⎝ i=1 ⎝ Q ⎠ ⎠ (5.5) 100 100 ⎛ ⎞ 100 S (s) f (s)ds p = S (s) f (s)ds. = S (s)dF(s). ∑∑⎜ ∫ t−1 ⎟ ij ∑ ∫ t−1 ∫ t−1 j=1 i=1 ⎝ Qi ⎠ i=1 Qi which is the total inheritance of all sons – that is, total wealth. 22 Thus, in the sons’ generation, there is, initially, heterogeneity of wealth at any skill level s′ . We now drop the assumption that all fathers die at the beginning of the year, and suppose, instead, that a fraction q of all fathers die at each skill level at the beginning of each year. Thus the capital at end of date t at skill level s will be an average of the capital of those who did not die and of the newly inheriting sons who arrive at skill level s. The average capital at skill level s at the end of the year is: * St (s) = (1 − q)St−1(s) + qSt−1(s) + It (s) . (5.6) The quantity in the first part of this convex combination is the average wealth of those who * do not die at the beginning of date t -- call it ‘survivors’ capital’-- and St−1(s) is the average wealth of sons who join type s at the beginning of date t. However, we do not attempt to keep track of the heterogeneity of wealth at each skill level that occurs as a result of death and inheritance. We only track average wealth at each skill level. Thus, we aggregate at each skill level s at each date, and assign everyone of that skill level the average amount of capital from (5.6). If you are the son of a wealthy father, your inheritance will add wealth to the cohort at your skill level, but it will not benefit you especially. The equations summarizing the model are the same 13 equations given at the end of subsection 4.A., except that we substitute equation (5.6) for (2.6) for the intergenerational transmission of wealth. We add the following equation to this set: 14. Wealth of an inheriting son, at the beginning of date t, whose own skill level is s′ : 100 100 p S (s) f (s)ds = S* (s′), for s′ ∈Q j , j = 1,...,100 (5.7) ∑ ij ∫ t−1 t−1 i=1 Qi As for solving the model, we proceed as previously up to and including the solving of * consumption and investment. St−1(⋅) is then determined by (5.7), and finally, St (⋅) is determined by (5.6). 23 B. Numerical results We have performed two sets of computations, one where 2% of the population dies and is replaced at the end of each period (q=0.02) and one where 4% get replaced (q=0.04). The results we obtain are all monotone in q, so we only report those with q=0.04. Insert Table 5 around here Compared with Model 3, the introduction of social mobility increases the equilibrium interest rate (from 5.7% to 7% with q=0.04) and decreases the equilibrium wage rate (from $71,461 to $64,942). The share of wealthy agents lobbying increases from the top 13.4% of capital owners to 21%, but the per capita lobbying contribution decreases (from $3,491 or 2.36% of GDP to $2,153 or 1.68% of GDP), resulting in an increase in the capital income tax (from 12.5% to 16.5%). GDP decreases by 14% (from $147,965 per capita to $128,185), and the capital/output ratio decreases from 5.05 to 3.96, while the share of capital income decreases from 38.7% to 35.7%. Social mobility is then detrimental to capital accumulation and to GDP, and results in less lobbying by wealthy agents. Its impact on the distribution of wealth is sizeable. First of all, the bottom half of the distribution starts accumulating a little bit of wealth (1.15% of total capital) for the first time in our simulations. Second, the patrimonial middle-class (with a wealth share increasing from 9.9% to 22.2%) and the 90th to 99th percentiles (with a share increasing from 36.2% to 40.3%) also benefit from social mobility. Finally, the losers are the top wealth owners, whose share decreases in a sizeable way (from 53.9% to 36.5% for the top 1%, and from 28.5% to 17% for the top 0.1%). Social mobility has then a detrimental impact on production (a large decrease in GDP, and an even larger decrease in capital accumulation), but improves wealth inequality. The share of wealth owned by the top 1% and the top .1% decrease significantly, to the benefit of all others. The bottom half of the distribution accumulates some (modest) wealth, for the first time in our simulations, but the main winner is the patrimonial middle class, whose wealth 24 share more than doubles. We obtain the exact same qualitative effects with the hybrid (lognormal-Pareto) skill distribution described in the Appendix. To conclude, we provide a comparison of some results obtained with the four models, first in Table 6 and then in Figures 6 to 8. Insert Table 6 around here Table 6 reports the consumption levels of individuals at the median and the 99.5th percentile of the skill/income distribution, as well as the percentage of GDP dedicated to lobbying activities by the top wealth earners, the Gini coefficient of the fitted income distribution, and the value of the demogrant. The median individual attains his highest consumption level with Model 1. His consumption level decreases with the introduction of lobbying and of differentiated rates of return on capital, but increases with social mobility. The pattern is opposite at the 99.5th percentile, with consumption increasing with lobbying and with differentiated rates of return, but decreasing with mobility. A similar pattern holds for lobbying expenses and for the Gini coefficients, which are maximal in Model 3. We then obtain a very clear picture, with lobbying and differentiated rates of return on capital benefiting the very rich, at the expense of the median, while mobility works in the opposite direction. The value of the demogrant decreases with the introduction of lobbying and of differentiated rates of return on capital, but also with the introduction of mobility. The latter effect may be counter-intuitive, because of the larger capital income tax rate in Model 4, and is due to the large decrease in capital accumulation (the share of total income devoted to the demogrant actually increases when we introduce social mobility). Social mobility then generates a trade-off between the consumption levels of the poorest (whose main source of 9 income is the demogrant) and of the median agent. 9 Once more, we obtain similar qualitative results with the hybrid skill distribution introduced in the Appendix. 25 Insert Figures 6, 7 and 8 around here Figure 6 depicts the absolute amount of wealth S as a function of skill s at t=25 for the 4 models studied, while Figure 7 depicts the distribution of wealth (the vertical dashed lines correspond to the 50th, 90th and 99th percentiles of the skill distribution). Both figures tell a similar story. Figures 6 and 7 show that there is little difference between the first two models in terms of absolute individual wealth and thus also of wealth distribution. Figure 6 shows that the top 10% of the skill distribution accumulates more capital in absolute value in Model 2 (where lobbying is introduced), which results in a shift, in the capital distribution, from the patrimonial middle class to the top 10%. Figure 6 also shows that the introduction of differentiated returns to capital induces the top 1.5% to accumulate more capital in absolute value, while the others accumulate less. This move is confirmed by looking at Figure 7. Moreover, Figure 7 shows that the impact of moving to Model 3 on the wealth distribution is sizeable. Finally, allowing for social mobility reduces very significantly the capital accumulation (in absolute value) of the top 10% of the skill distribution. In terms of distribution, this benefits not only the bottom half of the distribution, but also the patrimonial 10 middle class. Figure 8 shows the fitted Lorenz curves of the income distributions generated by the four models. The only Lorenz domination relationship is between Models 1 and 2, as stated above. 10 The discontinuities associated with Model 4 on Figures 6 and 7 are due to the fact that we use a 100 by 100 mobility matrix (with each row/column representing a centile of the income distribution) rather than a smooth function. 26 6. Conclusions In all models studied, the growth rates of income and of average wealth are significantly smaller than the interest rate (g In the first model studied, we observe a more unequal distribution of wealth as time passes, with the proportion of agents with no wealth increasing, and more generally, the 90th to 99th percentile of the wealth distribution gain shares at the expense of the bottom 90 percent, with the shares of the top 1% and the top 0.1% remaining roughly constant. The introduction of lobbying makes little difference with our calibration, although it decreases the capital income tax rate significantly and makes both the wealth and income distributions more unequal. Differentiated rates of return on capital and social mobility do affect the results much more, and in opposite directions. Differentiated returns benefit the top 1% and the top 0.1% of the wealth distribution, at the expense of all the others, and results in an increase of the Gini coefficient of the income distribution. Social mobility induces much less capital accumulation, but the shares of the top 1% and the top 0.1% decrease significantly to the benefit of all others, including those in the bottom half of the distribution, who have positive wealth uniquely in this scenario. The Gini coefficient of the income distribution decreases when social mobility is introduced. This improvement in wealth and income distributions is obtained at the price of a much lower GDP and overall capital accumulation, which in turn hurts the very poorest because of a decrease in the demogrant. All these results continue to hold when we model a hybrid lognormal-Pareto skill distribution, rather than a purely lognormal one. The only differences are that we obtain a better fit to the Jantzen and Volpert (2012)’s Lorenz curve with the hybrid skill distribution, and that the Gini coefficients are larger than with the pure lognormal skill distribution, whatever the model considered. [Table 7 about here] 27 Table 7 presents the Saez and Zucman (2014) top US wealth shares in 2012, which are virtually identical to the wealth shares in our four models at date t = 1 , and then presents the top wealth shares at date t = 25 from our four models. It seems that the most significant difference between our simulation results and Piketty’s history of recent capital accumulation is that in Model 4, when intergenerational mobility is introduced, the share of the top 1% and the top 0.1% of wealth-holders falls, albeit quite slowly. We think there are several possible explanations. The first is that ours is a competitive model, except for the introduction of variable rates of return in Models 3 and 4. We do note the rapid rate of increase of the wealth share of the top 1% and 0.1% in Model 3. In reality, the last thirty years in the United States may be best characterized as a period where capital has won significant gains through changing the bargaining conditions with labor, principally through the ascendency of neoliberal ideology and the concomitant attack on labor unions. One piece of evidence is that earnings have not increased with productivity. Another is the observed increase in the share of capital income. According to Piketty (2014, Figure 6.5), this share rose from 21% in 1975 to 28% in 2010; we do not see a comparable increase in our simulations, and this may be due to our competitive model. (In our first three models, capital’s share is remarkably stable over the 25 year period, and in the fourth model it falls a little bit.) If indeed the increase of the wealth share at the very top of the distribution is due to the ascendency of neoliberalism it may have been a one-shot phenomenon, and we might expect the wealth shares of the top 1% and 0.1% to become more stable, as they are in our simulations. This view is not inconsistent with one held by Piketty, that the share of income (if not wealth) at the top in the US has been due to the setting of executive compensation in a non-competitive way. Indeed, the high salaries of top managers may simply be a transfer from workers facilitated by neoliberalism. A second possible explanation has to do with the intergenerational transmission matrix of Chetty et al (2014) that we used. We employed this matrix to generate the intergenerational transmission of skills (which is to say labor income), while in reality it is a transmission matrix of all income of fathers to sons. There is also less persistence in wealth holding across generations in our model than there is in reality because we distribute all inherited 28 wealth among the entire skill cohort that a son enters, rather than keeping track of dynasties. Thus, our model implements more mixing of wealth than in fact occurs. In the other direction, we have assumed less dispersion of wealth than may indeed occur, if wealthy families have more than one child who inherits. It seems that a more nuanced approach to intergenerational mobility than ours is required. 29 Appendix: Introducing a Pareto tail in the skill distribution We assume that the skill distribution is lognormal (with cdf F and density f) up to skill x*, and then follows a Pareto distribution for x > x* . The CDF of the Pareto distribution is a P(x) = 1− ax−1/η , with corresponding density x−(1+η)/η . Jones (2015) reports that η = 0.6 η in the U.S. We then have two parameters to estimate: the threshold x* and a. In order to obtain a cut-off skill x* that is not too large, we have to truncate the domain of the distribution function to [0,B], where we choose (arbitrary) that B = 6. (Recall that our lognormal distribution has a mean of one and a median of 0.85.) We want the two density functions (lognormal and Pareto) to take the same value at x*, so that the overall density function is continuous over the whole range. Let b equal the integral of the distribution function that we construct on [0,B] before normalizing it to be 1. The normalized density on the first part is fx()/ b . We then solve a fx(*)/ b= x *−(1+ηη )/ , η from which we obtain ηfx()* a = . bx()*(1)/− +ηη We then solve the equation: B a Fx()/*(1)/ b+=∫ x− +ηη dx 1 x* η for b, using the formula for a. We obtain that b=1.0082 , a=0.193 and x*=3.2, with F(x* )/b = 0.982. In words, the distribution of skills is lognormal for the bottom 98.2% of the skill distribution, and Pareto for the top 1.8%. Table 8 below reports the numerical results obtained under Model 1 with this hybrid 11 distribution of skills. 11 Results obtained with the hybrid Lognormal-Pareto skill distribution for the four models are so similar to those obtained with the lognormal distribution that we do not report them here. They are available upon request from the authors. 30 Insert Table 8 here References Atkinson, A., T. Piketty and E. Saez, 2011. “Top incomes in the long run of history,” J. Econ. Lit. 49, 3-71 Bowles, S. and A. Jayadev, 2014. “One nation under guard,” New York Times, February 15. Chetty, R., N. Hendren, P. Kline and E. Saez, 2014. “Where is the land of opportunity? The geography of intergenerational mobility in the United States,” http://www.nber.org/papers/w19843 De Donder, Ph. and J. Roemer, 2013. “An allegory of the political influence of the top 1%”, CEPR DP9745 & CESifo WP 2013/4478. Jantzen, R. and K. Volpert, 2012. “On the Mathematics of Income Inequality: Splitting the Gini Index in Two”, The American Mathematical Monthly, 119 (10), 824-837 Jones, C., 2015. “Pareto and Piketty: The Macroeconomics of Top Income and Wealth Inequality”, Journal of Economic Perspectives, 29(1), 29–46 Karabarbounis, L. and B. Neiman, 2014. “The global decline of the labor share”, Quarterly Journal of Economics, 61-103. Piketty, T. 2014. Capital in the 21st century, Harvard University Press Piketty, T. and G. Zucman, 2013. “Capital is back: Wealth-income ratios in rich countries 1700-2010,” http://www.parisschoolofeconomics.com/zucman- gabriel/capitalisback/PikettyZucman2013WP.pdf Roemer, J. 2006. “Party competition under private and public financing: A comparison of institutions,” Advances in theoretical economics 6, Issue 1, article 2, http://www.bepress.com/bejte/advances/vol6/iss1/art2 Roemer, J. 2014. “Kantian optimization: A microfoundation for cooperation,” J. Public Econ. http://dx.doi.org/10.1016/j.jpubeco.2014.03.011 31 Saez E. and G. Zucman, 2014, “Wealth Inequality in the United States since 1913, evidence from capitalized income tax data”, NBER Working Paper 20265. Date 2015, 3, 16, 13, 49, 40.9174254 2 Figures and Tables.nb Model 1 : benchmark Table 1 : Equilibrium results as a function of t for Model 1 t r w GDP Kt K r d K wL shareK, 0 50 shareK, 50 90 shareK,90 99 shareK top 1 share top 0.1 GDP GDP GDP 1 0.0626065 68.2403 108.742 490.3 4.50883 0.372459 0.627541 0. 0.221455 0.357878 0.420668 0.221095 2 0.0625394 68.2732 109.909 496.166 4.51433 0.37261 0.62739 0. 0.215739 0.361377 0.422884 0.221951 3 0.0624353 68.3244 111.133 502.643 4.52289 0.372845 0.627155 0. 0.210715 0.364561 0.424725 0.222611 4 0.0623014 68.3905 112.408 509.647 4.53393 0.373148 0.626852 0. 0.206269 0.36748 0.426251 0.223105 5 0.0621435 68.4688 113.727 517.116 4.54701 0.373507 0.626493 0. 0.202315 0.370173 0.427512 0.223461 6 0.061966 68.5572 115.086 524.999 4.56178 0.373911 0.626089 0. 0.198798 0.372705 0.428581 0.223717 7 0.0617755 68.6527 116.48 533.215 4.57774 0.374347 0.625653 0. 0.195608 0.375008 0.429385 0.223836 8 0.0615663 68.7581 117.916 541.866 4.59536 0.374826 0.625174 0. 0.192749 0.377197 0.430055 0.223886 9 0.0613493 68.8681 119.381 550.795 4.61375 0.375326 0.624674 0. 0.190162 0.37926 0.430578 0.22386 10 0.0611237 68.9832 120.877 560.026 4.63301 0.375847 0.624153 0. 0.187817 0.381211 0.430973 0.223769 11 0.0608911 69.1028 122.404 569.545 4.65301 0.376387 0.623613 0. 0.185684 0.383063 0.431254 0.22362 12 0.0606528 69.2261 123.959 579.34 4.67365 0.376943 0.623057 0. 0.183739 0.384826 0.431435 0.223421 13 0.0604099 69.3526 125.542 589.399 4.69484 0.377512 0.622488 0. 0.181963 0.386511 0.431526 0.223177 14 0.0601634 69.482 127.152 599.715 4.71651 0.378092 0.621908 0. 0.180338 0.388125 0.431538 0.222892 15 0.0599141 69.6139 128.79 610.282 4.7386 0.378681 0.621319 0. 0.178849 0.389673 0.431478 0.222573 16 0.0596627 69.7479 130.453 621.094 4.76105 0.379278 0.620722 0. 0.177484 0.391164 0.431352 0.222221 17 0.0594098 69.8837 132.143 632.147 4.78382 0.379882 0.620118 0. 0.176231 0.392601 0.431169 0.221841 18 0.0591557 70.0212 133.858 643.438 4.80686 0.380491 0.619509 0. 0.17508 0.393989 0.430932 0.221436 19 0.0589011 70.1601 135.599 654.965 4.83015 0.381104 0.618896 0. 0.174021 0.395328 0.430643 0.221005 20 0.058646 70.3004 137.366 666.731 4.85367 0.381722 0.618278 0. 0.173051 0.396632 0.430317 0.220557 21 0.0583914 70.4414 139.158 678.719 4.87734 0.382341 0.617659 0. 0.172159 0.397895 0.429947 0.220089 22 0.0581368 70.5836 140.975 690.944 4.90119 0.382963 0.617037 0. 0.17134 0.399121 0.42954 0.219603 23 0.0578829 70.7265 142.817 703.401 4.92518 0.383587 0.616413 0. 0.170588 0.400314 0.429098 0.219101 24 0.0576297 70.8702 144.685 716.09 4.9493 0.384212 0.615788 0. 0.1699 0.401476 0.428625 0.218585 25 0.0573775 71.0144 146.578 729.01 4.97352 0.384838 0.615162 0. 0.16927 0.402608 0.428123 0.218056 Figures and Tables.nb 3 Figure 1 : Wealth as a function of skill s for the model 1 at t 25 S t 0 15 000 t 25 10 000 5000 s 0.5 1.0 1.5 2.0 2.5 3.0 3.5 4 Figures and Tables.nb Figure 2 : Wealth distribution as a function of skill for model 1 at t 25 capital share 0.5 0.4 0.3 0.2 0.1 s 0.5 1.0 1.5 2.0 2.5 3.0 3.5 Figures and Tables.nb 5 Figure 3 : Lorenz curves at t 25, obtained dots, fitted with lognormal blue curve and hybrid lognormal Pareto red curve, for Model 1 1.0 0.8 0.6 0.4 0.2 0.2 0.4 0.6 0.8 1.0 6 Figures and Tables.nb Model 2 (with lobbying) Table 2 : Equilibrium results as a function of t for Model 2 tr w1 Fs Σbar Τc GDP Kt K r d K wL shareK, shareK, shareK,90 shareK ShareK, GDP GDP GDP 0 50 50 90 99 top 1 top 0.1 1 0.0626065 68.2403 0.173762 1.48856 0.174127 108.742 490.3 4.50883 0.372459 0.627541 0. 0.219816 0.358513 0.42167 0.221634 2 0.0625343 68.2757 0.174368 1.53842 0.172503 109.915 496.24 4.51475 0.372622 0.627378 0. 0.212726 0.36253 0.424748 0.222954 3 0.062412 68.3359 0.174694 1.58667 0.17101 111.161 502.983 4.5248 0.372898 0.627102 0. 0.206533 0.366133 0.427334 0.224017 4 0.0622498 68.4161 0.174811 1.63368 0.169627 112.47 510.412 4.53819 0.373265 0.626735 0. 0.201087 0.369396 0.429517 0.224867 5 0.0620562 68.5123 0.174777 1.67979 0.168332 113.835 518.434 4.55427 0.373706 0.626294 0. 0.196272 0.372386 0.431375 0.225548 6 0.0618384 68.6211 0.174638 1.72531 0.167108 115.247 526.963 4.57246 0.374203 0.625797 0. 0.191972 0.375108 0.432919 0.226064 7 0.0615981 68.742 0.174405 1.77021 0.165953 116.708 535.999 4.59267 0.374753 0.625247 0. 0.18813 0.377637 0.434233 0.226462 8 0.0613428 68.8714 0.174121 1.81488 0.164847 118.208 545.446 4.6143 0.375341 0.624659 0. 0.184674 0.379988 0.435338 0.226751 9 0.0610745 69.0084 0.173798 1.85938 0.163787 119.746 555.291 4.63723 0.375961 0.624039 0. 0.181554 0.382185 0.436261 0.226947 10 0.0607958 69.152 0.173447 1.90384 0.162765 121.321 565.508 4.66125 0.376609 0.623391 0. 0.178725 0.384248 0.437027 0.227062 11 0.0605088 69.301 0.173082 1.94831 0.161779 122.931 576.077 4.68619 0.37728 0.62272 0. 0.176152 0.386193 0.437655 0.227106 12 0.0602154 69.4547 0.172709 1.99289 0.160822 124.573 586.981 4.71193 0.377969 0.622031 0. 0.173804 0.388034 0.438162 0.227089 13 0.0599169 69.6124 0.172335 2.03766 0.159892 126.248 598.206 4.73835 0.378674 0.621326 0. 0.171657 0.389782 0.438562 0.227018 14 0.0596147 69.7736 0.171965 2.08267 0.158985 127.953 609.744 4.76536 0.379392 0.620608 0. 0.169687 0.391448 0.438865 0.226898 15 0.0593097 69.9378 0.171602 2.12796 0.158099 129.689 621.585 4.79288 0.380121 0.619879 0. 0.167877 0.39304 0.439084 0.226736 16 0.0590027 70.1046 0.17125 2.17359 0.157232 131.455 633.723 4.82084 0.380859 0.619141 0. 0.16621 0.394565 0.439225 0.226536 17 0.0586945 70.2736 0.170909 2.21959 0.156383 133.25 646.154 4.84918 0.381604 0.618396 0. 0.164673 0.39603 0.439297 0.226302 18 0.0583858 70.4445 0.170583 2.26601 0.155549 135.074 658.872 4.87786 0.382355 0.617645 0. 0.163253 0.397438 0.439305 0.226037 19 0.0580768 70.6172 0.17027 2.3128 0.154731 136.927 671.878 4.90684 0.383111 0.616889 0. 0.161941 0.3988 0.439261 0.225746 20 0.0577686 70.7912 0.169975 2.3601 0.153925 138.807 685.159 4.93604 0.383869 0.616131 0. 0.160726 0.400114 0.439162 0.225429 21 0.0574608 70.9666 0.169695 2.40788 0.153132 140.717 698.727 4.96549 0.384631 0.615369 0. 0.159601 0.401385 0.439015 0.225089 22 0.0571541 71.1431 0.169431 2.45691 0.152327 142.654 712.577 4.99513 0.385395 0.614605 0. 0.158556 0.402614 0.438823 0.224728 23 0.0568486 71.3206 0.169182 2.50502 0.151579 144.621 726.712 5.02495 0.38616 0.61384 0. 0.15759 0.403813 0.438597 0.224351 24 0.056545 71.4987 0.168952 2.55433 0.15082 146.614 741.116 5.05487 0.386925 0.613075 0. 0.156693 0.404976 0.438331 0.223955 25 0.056243 71.6776 0.168737 2.60426 0.150069 148.637 755.807 5.08493 0.387691 0.612309 0. 0.155861 0.406107 0.438032 0.223545 Figures and Tables.nb 7 Model 3 (with lobbying and differential rates of return) Table 3 : Equilibrium results as a function of t for Model 3 K wL tr Ν w1 Fs Σbar Τc GDP Kt r d K shareK, shareK, shareK,90 shareK ShareK, top GDP GDP GDP 0 50 50 90 99 top 1 0.1 1 0.0626065 1.51974 68.2403 0.173762 1.98082 0.152835 108.742 490.3 4.50883 0.372459 0.627541 0. 0.216021 0.357999 0.42598 0.224161 2 0.0625959 1.41893 68.2455 0.172823 2.04037 0.151278 109.842 495.356 4.5097 0.372483 0.627517 0. 0.205307 0.361317 0.433377 0.228027 3 0.0625283 1.32704 68.2787 0.171574 2.09914 0.149806 111.022 501.29 4.51524 0.372635 0.627365 0. 0.195674 0.36405 0.440277 0.231643 4 0.0624141 1.24239 68.3349 0.170101 2.15743 0.148405 112.27 507.981 4.52463 0.372893 0.627107 0. 0.186966 0.366281 0.44675 0.235046 5 0.0622618 1.16369 68.4101 0.168468 2.21554 0.14706 113.58 515.338 4.5372 0.373238 0.626762 0. 0.179059 0.368081 0.45286 0.238267 6 0.0620784 1.08986 68.5012 0.166726 2.2735 0.145767 114.945 523.279 4.55241 0.373655 0.626345 0. 0.171843 0.369505 0.458652 0.241331 7 0.0618693 1.02016 68.6056 0.16491 2.33178 0.144508 116.36 531.751 4.56986 0.374132 0.625868 0. 0.16523 0.370603 0.464168 0.244258 8 0.0616393 0.953905 68.7212 0.163051 2.39006 0.143293 117.821 540.705 4.5892 0.374659 0.625341 0. 0.159146 0.371413 0.469441 0.247068 9 0.0613919 0.890605 68.8464 0.16117 2.44886 0.142103 119.325 550.102 4.61013 0.375227 0.624773 0. 0.153527 0.371972 0.474501 0.249773 10 0.0611305 0.829811 68.9797 0.159282 2.50815 0.140938 120.868 559.913 4.63243 0.375831 0.624169 0. 0.14832 0.372308 0.479372 0.252388 11 0.0608576 0.771176 69.12 0.157401 2.56786 0.139799 122.449 570.112 4.6559 0.376465 0.623535 0. 0.14348 0.372446 0.484074 0.254922 12 0.0605754 0.714417 69.2663 0.155537 2.62827 0.138678 124.067 580.68 4.68038 0.377124 0.622876 0. 0.138967 0.372409 0.488625 0.257384 13 0.0602857 0.65928 69.4177 0.153695 2.68937 0.137573 125.719 591.598 4.70574 0.377804 0.622196 0. 0.134748 0.372214 0.493038 0.259781 14 0.0599901 0.605582 69.5736 0.151881 2.75122 0.136484 127.404 602.856 4.73185 0.378501 0.621499 0. 0.130794 0.371877 0.497328 0.262121 15 0.0596897 0.553131 69.7335 0.150098 2.81382 0.13541 129.122 614.442 4.75863 0.379214 0.620786 0. 0.127081 0.371415 0.501505 0.264409 16 0.0593858 0.501774 69.8967 0.14835 2.87725 0.134348 130.871 626.345 4.78598 0.379939 0.620061 0. 0.123584 0.370838 0.50558 0.26665 17 0.0590793 0.451393 70.0628 0.146638 2.94161 0.133295 132.651 638.559 4.81383 0.380675 0.619325 0. 0.120286 0.370156 0.509559 0.268847 18 0.058771 0.401888 70.2315 0.144962 3.00683 0.132254 134.462 651.08 4.84212 0.381419 0.618581 0. 0.11717 0.369381 0.51345 0.271006 19 0.0584615 0.353155 70.4025 0.143324 3.07298 0.131223 136.303 663.903 4.8708 0.38217 0.61783 0. 0.11422 0.36852 0.517261 0.273128 20 0.0581514 0.305101 70.5754 0.141723 3.1401 0.130201 138.173 677.023 4.89982 0.382928 0.617072 0. 0.111423 0.367581 0.520996 0.275217 21 0.0578413 0.257676 70.75 0.140159 3.2082 0.129187 140.073 690.438 4.92913 0.38369 0.61631 0. 0.108766 0.366571 0.524662 0.277276 22 0.0575316 0.210786 70.9261 0.138632 3.27729 0.128182 142.002 704.146 4.95869 0.384455 0.615545 0. 0.106241 0.365497 0.528262 0.279307 23 0.0572226 0.164383 71.1035 0.137141 3.34744 0.127183 143.961 718.146 4.98848 0.385224 0.614776 0. 0.103836 0.364363 0.5318 0.281312 24 0.0569147 0.118422 71.282 0.135685 3.41863 0.126192 145.948 732.436 5.01846 0.385994 0.614006 0. 0.101544 0.363175 0.535281 0.283292 25 0.0566082 0.0728587 71.4615 0.134263 3.49095 0.125207 147.965 747.018 5.04862 0.386765 0.613235 0. 0.0993559 0.361937 0.538707 0.285249 8 Figures and Tables.nb Figure 4 : Rate of return on capital as a function of skill s in Model 3 rs,t 0.08 0.06 0.04 0.02 Fs 0.2 0.4 0.6 0.8 1.0 Figures and Tables.nb 9 Figure 5 : Lorenz curve at t 25, obtained dots and fitted curve, in Model 3 1.0 0.8 0.6 0.4 0.2 0.2 0.4 0.6 0.8 1.0 10 Figures and Tables.nb Model 4 (with lobbying, differential rates of return and mobility) Table Not in paper : Equilibrium results as a function of t for Model 4 with 2 death rate mobility K wL tr Ν w1 Fs Σbar Τc GDP Kt r d K shareK, shareK, shareK,90 shareK ShareK, GDP GDP GDP 0 50 50 90 99 top 1 top 0.1 1 0.0626065 1.51974 68.2403 0.173762 1.98082 0.152835 108.742 490.3 4.50883 0.372459 0.627541 0.00580602 0.220607 0.355472 0.418645 0.219796 2 0.0625772 1.51543 68.2546 0.173993 2.00878 0.152514 109.864 495.623 4.51123 0.372525 0.627475 0.00578709 0.215157 0.358473 0.421111 0.220548 3 0.062717 1.47575 68.1862 0.175 2.04074 0.152053 110.796 498.561 4.4998 0.37221 0.62779 0.00577091 0.210289 0.361173 0.423301 0.221157 4 0.0628315 1.44067 68.1303 0.175668 2.07221 0.151629 111.767 501.885 4.49047 0.371953 0.628047 0.00575685 0.20591 0.363604 0.425252 0.221642 5 0.0629244 1.40956 68.0851 0.176296 2.10324 0.151238 112.772 505.549 4.48293 0.371744 0.628256 0.00574466 0.201943 0.365824 0.427018 0.222034 6 0.0630014 1.38119 68.0478 0.176796 2.13396 0.150876 113.806 509.475 4.47669 0.371572 0.628428 0.00573397 0.198321 0.367864 0.428632 0.222348 7 0.0630643 1.35542 68.0173 0.177183 2.16462 0.150532 114.867 513.64 4.47161 0.371431 0.628569 0.00572428 0.194997 0.369725 0.43009 0.222583 8 0.0631144 1.33208 67.9931 0.177471 2.19513 0.150209 115.954 518.03 4.46757 0.371319 0.628681 0.00573009 0.191935 0.371456 0.431444 0.222767 9 0.0631542 1.31055 67.9738 0.177677 2.22568 0.1499 117.063 522.613 4.46436 0.37123 0.62877 0.00572226 0.189103 0.373051 0.432683 0.222893 10 0.0631851 1.29064 67.959 0.177814 2.25609 0.149609 118.195 527.372 4.46187 0.371161 0.628839 0.00571525 0.186475 0.374535 0.433833 0.222976 11 0.0631966 1.27547 67.9534 0.17783 2.28652 0.149333 119.363 532.47 4.46095 0.371136 0.628864 0.0056943 0.184018 0.375928 0.434907 0.223022 12 0.0632248 1.25491 67.9398 0.17792 2.31711 0.149065 120.52 537.359 4.45868 0.371073 0.628927 0.00568651 0.181666 0.377102 0.435765 0.22296 13 0.0632238 1.24212 67.9403 0.177842 2.34765 0.148812 121.726 542.749 4.45876 0.371075 0.628925 0.00568288 0.179589 0.378435 0.436833 0.223007 14 0.0632403 1.22381 67.9324 0.177846 2.37834 0.148567 122.922 547.918 4.45743 0.371038 0.628962 0.00567797 0.177594 0.379571 0.437705 0.222956 15 0.0632411 1.20952 67.932 0.177756 2.40922 0.14833 124.15 553.384 4.45737 0.371036 0.628964 0.00567348 0.175729 0.380632 0.438516 0.222878 16 0.0632365 1.19638 67.9342 0.177632 2.44007 0.148106 125.398 558.991 4.45774 0.371047 0.628953 0.00566913 0.173955 0.381635 0.439283 0.222778 17 0.0632289 1.18383 67.9378 0.177487 2.471 0.147892 126.662 564.703 4.45835 0.371063 0.628937 0.00566521 0.172294 0.382583 0.440008 0.22266 18 0.0632184 1.17193 67.9429 0.177321 2.50246 0.147673 127.943 570.524 4.4592 0.371087 0.628913 0.00566151 0.17072 0.383478 0.44069 0.222523 19 0.0632049 1.16061 67.9494 0.177138 2.53396 0.147466 129.241 576.451 4.46028 0.371117 0.628883 0.00565811 0.169236 0.384327 0.441338 0.22237 20 0.0631887 1.14992 67.9572 0.176939 2.56557 0.147268 130.556 582.486 4.46158 0.371153 0.628847 0.00565471 0.167828 0.385115 0.441932 0.222193 21 0.0631686 1.14019 67.9669 0.176718 2.59741 0.147075 131.89 588.65 4.4632 0.371198 0.628802 0.00565177 0.166491 0.385888 0.442522 0.222016 22 0.0631468 1.13075 67.9774 0.176489 2.62938 0.146889 133.24 594.908 4.46495 0.371247 0.628753 0.00564892 0.165223 0.386613 0.443071 0.221821 23 0.0631268 1.12084 67.9871 0.17627 2.66178 0.146702 134.601 601.203 4.46657 0.371291 0.628709 0.00564694 0.164043 0.38735 0.443645 0.22164 24 0.0631079 1.11052 67.9962 0.176058 2.69414 0.146529 135.974 607.544 4.46809 0.371334 0.628666 0.00564436 0.162896 0.387997 0.444132 0.221419 25 0.0630815 1.10229 68.009 0.175807 2.72713 0.146348 137.372 614.084 4.47022 0.371392 0.628608 0.00564224 0.161826 0.388634 0.444617 0.221198 Figures and Tables.nb 11 Table 5 : Equilibrium results as a function of t for Model 4 with 4 death rate mobility K r d K wL tr Ν w1 Fs Σbar Τc GDP Kt shareK, 0 50 shareK, shareK,90 99 shareK top 1 ShareK, GDP GDP GDP 50 90 top 0.1 1 0.0626065 1.51974 68.2403 0.173762 1.98082 0.152835 108.742 490.3 4.50883 0.372459 0.627541 0.011612 0.225194 0.352945 0.411311 0.215432 2 0.062559 1.61157 68.2636 0.175 1.97703 0.153769 109.886 495.884 4.51272 0.372566 0.627434 0.0116006 0.22496 0.35557 0.408928 0.213135 3 0.0629044 1.62188 68.0949 0.178383 1.98277 0.154321 110.574 495.873 4.48455 0.371789 0.628211 0.0115908 0.224853 0.358081 0.406527 0.210844 4 0.0632437 1.63273 67.9307 0.181723 1.98849 0.154874 111.276 495.975 4.45716 0.371031 0.628969 0.0115827 0.224836 0.360505 0.404134 0.208571 5 0.0635768 1.64431 67.7711 0.184992 1.99445 0.155417 111.993 496.188 4.43054 0.37029 0.62971 0.0115758 0.224889 0.362846 0.401753 0.206319 6 0.0639057 1.65608 67.615 0.187846 2.00064 0.155949 112.722 496.486 4.40451 0.369564 0.630436 0.0115697 0.224973 0.365114 0.399393 0.204093 7 0.0642316 1.66792 67.4618 0.190914 2.00566 0.156526 113.462 496.849 4.37898 0.368848 0.631152 0.0115642 0.22507 0.367337 0.397082 0.201905 8 0.0645549 1.67982 67.3111 0.193914 2.01337 0.156998 114.213 497.273 4.35389 0.368142 0.631858 0.0115591 0.225156 0.369513 0.394816 0.199755 9 0.0648773 1.69136 67.1623 0.195 2.01985 0.157518 114.973 497.731 4.3291 0.367443 0.632557 0.0115556 0.225251 0.37168 0.392636 0.197662 10 0.0651994 1.70236 67.0149 0.195535 2.0268 0.158019 115.741 498.215 4.30458 0.366747 0.633253 0.0115506 0.225306 0.373753 0.390447 0.195578 11 0.0655208 1.71306 66.8691 0.19787 2.03379 0.158518 116.517 498.732 4.28033 0.366058 0.633942 0.0115464 0.225342 0.375813 0.388335 0.193546 12 0.065841 1.72362 66.7252 0.20013 2.04098 0.159009 117.302 499.287 4.2564 0.365374 0.634626 0.0115425 0.225336 0.37786 0.386298 0.191565 13 0.0661625 1.73351 66.5819 0.202327 2.04855 0.159486 118.094 499.846 4.2326 0.364692 0.635308 0.0115392 0.225313 0.379895 0.384333 0.189633 14 0.0664854 1.74262 66.4393 0.204248 2.05876 0.159865 118.892 500.408 4.20892 0.36401 0.63599 0.0115353 0.22524 0.381879 0.382403 0.18773 15 0.066806 1.75192 66.299 0.205501 2.06428 0.160417 119.701 501.023 4.18563 0.363338 0.636662 0.0115325 0.225135 0.383849 0.380541 0.185874 16 0.067128 1.76049 66.1593 0.206488 2.07205 0.160886 120.516 501.641 4.16245 0.362666 0.637334 0.0115289 0.224983 0.385804 0.378744 0.184062 17 0.0674509 1.76856 66.0204 0.20735 2.08043 0.161333 121.338 502.268 4.13942 0.361996 0.638004 0.0115183 0.22466 0.387494 0.376769 0.182176 18 0.0677504 1.782 65.8926 0.208016 2.08858 0.161788 122.196 503.23 4.11824 0.361377 0.638623 0.0115206 0.224524 0.389623 0.37529 0.180542 19 0.0680977 1.78352 65.7457 0.208747 2.09749 0.162216 123.005 503.572 4.09391 0.360664 0.639336 0.0115179 0.224247 0.391562 0.373711 0.178872 20 0.0684244 1.78983 65.6087 0.209295 2.10588 0.162663 123.847 504.212 4.07124 0.359997 0.640003 0.0115134 0.223898 0.393424 0.372135 0.177214 21 0.0687483 1.79655 65.4741 0.209738 2.11469 0.163096 124.701 504.911 4.04898 0.35934 0.64066 0.0115104 0.223534 0.395322 0.370666 0.175618 22 0.0690775 1.80172 65.3385 0.210109 2.1242 0.163504 125.557 505.561 4.02655 0.358675 0.641325 0.0115067 0.223124 0.397172 0.369226 0.174047 23 0.0694045 1.80712 65.2049 0.210387 2.13335 0.163925 126.424 506.261 4.00448 0.358019 0.641981 0.0115035 0.222691 0.399017 0.367852 0.172519 24 0.0697336 1.81173 65.0716 0.210592 2.1428 0.164336 127.297 506.955 3.98247 0.357361 0.642639 0.0114988 0.222197 0.400788 0.366481 0.171002 25 0.0700572 1.81732 64.9417 0.210706 2.15257 0.164737 128.185 507.743 3.96103 0.356719 0.643281 0.0114964 0.221711 0.402621 0.365235 0.169554 12 Figures and Tables.nb Comparison of the 4 models Table 6 : Consumption and lobbying for the 4 models at t 25 Lobby Model cmedian,25 c0.995,25 Gini Demogrant GDP 1 96.018 583.739 0 0.38631 46.1994 2 88.5172 668.046 1.75 0.441436 38.2333 3 87.1849 899.987 2.36 0.468321 37.0527 4q4 95.5686 660.111 1.679 0.417872 34.7204 Figure 6 : S s, t for the 4 models at t 25 Ss,25 Model 1 Model 2 15 000 Model 3 Model 4 4 10 000 5000 s 0.5 1.0 1.5 2.0 2.5 3.0 3.5 Figures and Tables.nb 13 Figure 7 : Wealth distribution as a function of skill for the 4 models at t 25 capital share 0.5 Model 1 Model 2 Model 3 0.4 Model 4 4 0.3 0.2 0.1 s 0.5 1.0 1.5 2.0 2.5 3.0 3.5 14 Figures and Tables.nb Figure 8 : Lorenz curves for the 4 models at t 25 1.0 Model 1 Model 2 0.8 Model 3 Model 4 4 0.6 0.4 0.2 0.2 0.4 0.6 0.8 1.0 Figures and Tables.nb 15 Table 7 : Top wealth shares from Saez Zucman S Z in 2012 and in our four models 2037 Model Top decile Top centile Top 0.1 SZ 2012 77.2 41.8 22. 12037 82.8 42.8 21.8 22037 84.4 43.8 22.3 32037 90. 53.8 28.5 42037 76.7 36.3 17. Table 8 : Equilibrium results as a function of t for Model 1 with the Lognorma Pareto skill distribution t r w GDP Kt K r d K wL shareK, shareK, shareK,90 shareK share GDP GDP GDP 050 5090 99 top 1 top 0.1 1 0.0635969 67.7615 110.704 490.3 4.42894 0.370246 0.629754 0. 0.221665 0.357912 0.420455 0.220813 2 0.0634755 67.8195 111.954 496.92 4.43861 0.370515 0.629485 0. 0.216157 0.361447 0.422429 0.22139 3 0.0633179 67.895 113.263 504.155 4.4512 0.370865 0.629135 0. 0.211332 0.364669 0.424032 0.221774 4 0.0631316 67.9847 114.622 511.922 4.46618 0.371281 0.628719 0. 0.207079 0.367627 0.425326 0.221997 5 0.0629224 68.0861 116.027 520.16 4.48309 0.371749 0.628251 0. 0.20331 0.370362 0.426361 0.222086 6 0.062695 68.197 117.474 528.82 4.5016 0.37226 0.62774 0. 0.199952 0.372907 0.427174 0.222062 7 0.0624528 68.3158 118.959 537.865 4.52144 0.372806 0.627194 0. 0.196948 0.375286 0.427798 0.221942 8 0.062199 68.4413 120.48 547.267 4.5424 0.373381 0.626619 0. 0.19425 0.377522 0.42823 0.221741 9 0.0619359 68.5723 122.034 557. 4.5643 0.37398 0.62602 0. 0.191818 0.379634 0.428551 0.221468 10 0.0616654 68.7081 123.621 567.048 4.587 0.374599 0.625401 0. 0.189621 0.381634 0.428748 0.221135 11 0.061389 68.8479 125.238 577.395 4.61038 0.375234 0.624766 0. 0.18763 0.383536 0.428837 0.220749 12 0.0611082 68.9912 126.886 588.032 4.63434 0.375883 0.624117 0. 0.185822 0.38535 0.42883 0.220317 13 0.060824 69.1374 128.562 598.946 4.65881 0.376544 0.623456 0. 0.184178 0.387085 0.428739 0.219844 14 0.0605373 69.2861 130.267 610.133 4.68371 0.377213 0.622787 0. 0.182681 0.38875 0.428572 0.219336 15 0.0602489 69.4371 132. 621.584 4.70898 0.37789 0.62211 0. 0.181316 0.39035 0.428337 0.218796 16 0.0599594 69.5899 133.76 633.297 4.73457 0.378574 0.621426 0. 0.18007 0.39189 0.428042 0.218229 17 0.0596695 69.7443 135.548 645.267 4.76044 0.379262 0.620738 0. 0.178934 0.393377 0.427692 0.217637 18 0.0593794 69.9001 137.362 657.492 4.78656 0.379954 0.620046 0. 0.177896 0.394814 0.427292 0.217024 19 0.0590897 70.0571 139.203 669.97 4.81288 0.380649 0.619351 0. 0.176949 0.396206 0.426847 0.216391 20 0.0588007 70.2152 141.072 682.699 4.83938 0.381347 0.618653 0. 0.176086 0.397555 0.426362 0.215741 21 0.0585127 70.3741 142.966 695.68 4.86604 0.382046 0.617954 0. 0.175299 0.398864 0.425839 0.215075 22 0.0582258 70.5338 144.888 708.911 4.89283 0.382746 0.617254 0. 0.174584 0.400136 0.425282 0.214396 23 0.0579403 70.6941 146.836 722.393 4.91974 0.383446 0.616554 0. 0.173933 0.401372 0.424691 0.213703 24 0.0576563 70.855 148.811 736.131 4.94676 0.384147 0.615853 0. 0.173346 0.402579 0.424077 0.213001 25 0.0573743 71.0162 150.812 750.112 4.97382 0.384846 0.615154 0. 0.172815 0.403753 0.423433 0.212289